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This movie gives a brief overview of the strategy for choosing your row operations no matter
what it is you’re trying to do. Whether it’s to solve a system of equations or to
find an inverse.
Remember, there are three primary kinds of row operations. One of them is to multiply
a number times a row. The occasions when you would do that is when you are trying to produce
a one in a column. You just move to work in a new column and you need to get your one
in the proper spot. The second kind of row operation, where you’re adding a multiple
of a row to another row, is done for the purpose of creating a zero. Row-three minus three-row-one
would be to put a zero in row-three using a one that you had already created in row-one.
Finally the swapping of two rows is done primarily when you need a one at a spot where you presently
have a zero. You use the swap operation to swap out that row with the zero, to bring
in another row to that location where there is a non-zero number that you can work with
to turn into a one.
Here’s a generic situation where we have cleaned up the first column and we’re moving
on to the second column. One very important thing to keep in mind is you never have to
look beyond that. What’s sitting over here is completely irrelevant. In deciding what
row operations we’re going to do, all we have to look at are the numbers in this column.
You never have to look over beyond that in deciding your strategy. In this example we
have the one in column-two, we want the minus-six and the four to become zeros, and again remember
the row operations that are chosen it takes one row operation to do each. Row-one plus
six-row-two because this is a minus-six. Row-three minus four times row-two because this is a
plus four. These two row operations will do what you need. This one will put the zero
here. This one will put the zero in the third row.
How many row operations you need to do to clean up a column will, of course, depend
on how many rows you’re working with. In this instance we’re working with four rows.
We have the one in column-three. We need zeros in these other locations. It will take three
row operations in order to accomplish that. One row operation for each zero we’re trying
to produce.
Here’s an example where we have finished with column one, we have finished with column
two, we’re beginning to work on column three, we would like to have a one where the zero
is, we can’t produce a one there by multiplying row-three times something. What we do is we
swap row-three and row-four. If we swap those two rows, that will put minus-three in that
spot, and then we can turn that into a one by multiplying row-three times minus a third
after we’ve done that.